Inventory control in multi-channel retail

In recent years multi-channel retail systems have received increasing interest. Partly due to growing online business that serves as a second sales channel for many firms, offering channel specific prices has become a common form of revenue management. We analyze conditions for known inventory control policies to be optimal in presence of two different sales channels. We propose a single item lost sales model with a lead time of zero, periodic review and nonlinear non-stationary cost components without rationing to realistically represent a typical web-based retail scenario. We analyze three variants of the model with different arrival processes: demand not following any particular distribution, Poisson distributed demand and a batch arrival process where demand follows a Pòlya frequency type distribution. We show that without further assumptions on the arrival process, relatively strict conditions must be imposed on the penalty cost in order to achieve optimality of the base stock policy. We also show that for a Poisson arrival process with fixed ordering costs the model with two sales channels can be transformed into the well known model with a single channel where mild conditions yield optimality of an (s, S) policy. Conditions for optimality of the base stock and (s, S) policy for the batch arrival process with and without fixed ordering costs, respectively, are presented together with a proof that the batch arrival process provides valid upper and lower bounds for the optimal value function.     

Optimal scheduling of non-perfect inspections

^** Corresponding author. Email: romulo.zequeira{at}utt.fr^*** Email: christophe.berenguer{at}utt.fr In this paper, we study the determination of optimal inspectionpolicies when three types of inspections are available: partial,perfect and imperfect. Perfect inspections diagnose withouterror the system state. The system can fail because of threecompeting failure types: I, II and III. Partial inspectionsdetect without error type I failures. Failures of type II canbe detected by imperfect inspections which have non-zero probabilityof false positives. Partial and imperfect inspections are madeat the same time. Type III failures are detectable only by perfectinspections. If the system is found failed in an inspection,a repair is made which renders the system in a good-as-new condition.The system is preventively maintained following an age-basedpolicy. Preventive maintenance actions return the system toa good-as-new condition. We consider cost contributions of inspections,repairs, preventive maintenance and periods of unavailability.The model presented permits to determine the optimal (constant)inter-inspection period for partial, imperfect and perfect inspectionsand the optimal times of preventive maintenance actions.     

Non-parametric predictive reliability demonstration for failure-free periods

We derive sample sizes for testing as required for reliabilitydemonstration, using non-parametric predictive inference forthe predicted number of future failures on the basis of testinformation. We assume that tests lead to zero failures, asis typical, for example, for high-reliability testing. Optimizationis in terms of failure-free periods for a process after testing,and we also consider total expected cost minimization, withparticular attention to deterministic and Poisson processes.We show that, perhaps surprisingly, the deterministic case isa worst-case scenario with regard to number of tests needed.     

A reliability semi‐Markov model involving geometric processes

We consider a semi‐Markov process that models the repair and maintenance of a repairable system in steady state. The operating and repair times are independent random variables with general distributions. Failures can be caused by an external source or by an internal source. Some failures are repairable and others are not. After a repairable failure, the system is not as good as new and our model reflects that. At a non‐repairable failure, the system is replaced by a new one. We assume that external failures occur according to a Poisson process. Moreover, there is an upper limit N of repairs, it is replaced by a new system at the next failure, regardless of its type. Operational and repair times are affected by multiplicative rates, so they follow geometric processes. For this system, the stationary distribution and performance measures as well as the availability and the rate of occurrence of different types of failures in stationary state are calculated. Copyright © 2002 John Wiley & Sons, Ltd.     

A number-dependent replacement policy for a system with continuous preventive maintenance and random lead times

This paper considers a number-dependent replacement policy for a system with two failure types that is replaced at the nth type I (minor) failure or the first type II (catastrophic) failure, whichever occurs first. Repair or replacement times are instantaneous but spare/replacement unit delivery lead times are random. Type I failures are repaired at zero cost since preventive maintenance is performed continuously. Type II failures, however, require costly system replacement. A model is developed for the average cost per unit time based on the stochastic behavior of the system and replacement, storage, and downtime costs. The cost-minimizing policy is derived and discussed. We show that the optimal number of type I failures triggering replacement is unique under certain conditions. A numerical example is presented and a sensitivity analysis is performed.     

The effect of testing equipment shift on optimal decisions in a repetitive testing process

Repetitive testing process is commonly used in the final testing stage of semiconductor manufacturing to ensure high outgoing product quality and to reduce testing errors. The decision on testing lot size and the number of testing repetitions ultimately determines the effectiveness of the testing process. Setting the retest rule is often difficult in practice due to uncertainties in the incoming product quality and testing equipment condition. In this paper, we study a repetitive testing process where the testing equipment may shift randomly to an inferior state. We develop a cost model that helps us to make optimal decisions on retesting rule. Through numerical analysis, we provide practical insights about the effects of testing equipment shift rate, testing errors, and different costs such as cost of testing and cost of rejecting conforming products on the optimal decision and the system performance. We find that significant penalty may result if the potential testing equipment shift is ignored.     

Markov Decision Processes with Variance Minimization: A New Condition and Approach

Abstract

This article deals with the limiting average variance criterion for discrete-time Markov decision processes in Borel spaces. The costs may have neither upper nor lower bounds. We propose another set of conditions under which we prove the existence of a variance minimal policy in the class of average expected cost optimal stationary policies. Our conditions are weaker than those in the previous literature. Moreover, some sufficient conditions for the existence of a variance minimal policy are imposed on the primitive data of the model. In particular, the stochastic monotonicity condition in this paper has been first used to study the limiting average variance criterion. Also, the optimality inequality approach provided here is different from the “optimality equation approach” widely used in the previous literature. Finally, we use a controlled queueing system to illustrate our results.     

A disaster queue with Markovian arrivals and impatient customers

We consider a single server queueing system in which arrivals occur according to a Markovian arrival process. The system is subject to disastrous failures at which times all customers in the system are lost. Arrivals occurring during the time the system undergoes repair are stored in a buffer of finite capacity. These customers can become impatient after waiting a random amount of time and leave the system. However, these customers do not become impatient once the system becomes operable. When the system is operable, there is no limit on the number of customers who can be admitted. The structure of this queueing model is of GI/M/1-type that has been extensively studied by Neuts and others. The model is analyzed in steady state by exploiting the special nature of this type queueing model. A number of useful performance measures along with some illustrative examples are reported.     

Dual Sufficient Optimality Conditions for the Generalized Problem of Bolza

We develop sufficient conditions for optimality in the generalized problem of Bolza. The basis of our approach is the dual Hamilton–Jacobi inequality leading to a new sufficient criterion for optimality in which we assume the existence of a function satisfying, together with the Hamiltonian, a certain inequality. Consequently, using this criterion, we derive new sufficient conditions for optimality of first and second order for a relative minimum.     

Flexible servers in tandem lines with setup costs

We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. The goal is to maximize the long-run average profit. We provide a general problem formulation and some structural results, and then concentrate on tandem lines with two stations, two servers, and a finite buffer between the stations. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold” policies (where servers’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and various service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.     

Control of arrivals in a finite buffered queue with setup costs

We consider finite buffered queues where the arrival process is controlled by shutting down and restarting the arrival stream. In the absence of holding costs for items in the queue, the optimal (s,?S) policy can be characterised by relating the arrival control problem to a corresponding service control problem. With the inclusion of holding costs however, this characterisation is not valid and efficient numerical computation of the queue length probability distribution is necessary. We perform this computation by using a duality property which relates queue lengths in the controlled arrival system to a controlled service system. Numerical results which analyse the effect of setup and holding costs and the variability of the arrival process on the performance of the system are included.     

Optimal Pricing and Admission Control in a Queueing System with Periodically Varying Parameters

We consider congestion control in a nonstationary queueing system. Assuming that the arrival and service rates are bounded, periodic functions of time, a Markov decision process (MDP) formulation is developed. We show under the infinite horizon discounted and average reward optimality criteria, for each fixed time, optimal pricing and admission control strategies are nondecreasing in the number of customers in the system. This extends stationary results to the nonstationary setting. Despite this result, the problem still seems intractable. We propose an easily implementable pointwise stationary approximation (PSA) to approximate the optimal policies, suggest a heuristic to improve the implementation of the PSA and verify its usefulness via a numerical study.     

Single machine scheduling when processing times are correlated normal random variables

A single machine is available to process a collection of jobs whose processing times are jointly multivariate normal. Processing is nonpreemptive. We show that in the equicorrelation case, the permutation policy which schedules the jobs in ascending order of their (marginal) mean processing times is optimal for general order-specific costs and also for job-specific costs under an agreeability condition. Suboptimality bounds on the performance of Smith's rule are obtained for the weighted flow-time criterion. A computational study shows that a dynamic version of Smith's rule comes very close to optimality.     

A maintenance model with failures and inspection following Markovian arrival processes and two repair modes

We consider a deteriorating system submitted to external and internal failures, whose deterioration level is known by means of inspections. There are two types of repairs: minimal and perfect, depending on the deterioration level, each one following a different phase-type distribution. The failures and the inspections follow different Markovian arrival processes (MAP). Under these assumptions, the system is governed by a generalized Markov process, whose state space and generator are constructed. This general model includes the phase-type renewal process as a special case. The distribution of the number of minimal and perfect repairs between two inspections are determined. A numerical application optimizing costs is performed, and different particular cases of the model are compared.     

On the bailout dividend problem with periodic dividend payments for spectrally negative Markov additive processes

This paper studies the bailout optimal dividend problem with regime switching under the constraint that dividend payments can be made only at the arrival times of an independent Poisson process while capital can be injected continuously in time. We show the optimality of the regime-modulated Parisian-classical reflection strategy when the underlying risk model follows a general spectrally negative Markov additive process. In order to verify the optimality, first we study an auxiliary problem driven by a single spectrally negative Lévy process with a final payoff at an exponential terminal time and characterize the optimal dividend strategy. Then, we use the dynamic programming principle to transform the global regime-switching problem into an equivalent local optimization problem with a final payoff up to the first regime switching time. The optimality of the regime modulated Parisian-classical barrier strategy can be proven by using the results from the auxiliary problem and approximations via recursive iterations.     

Optimal batch service of a polling system under partial information

We consider the optimal scheduling of an infinite-capacity batch server in aN-node ring queueing network, where the controller observes only the length of the queue at which the server is located. For a cost criterion that includes linear holding costs, fixed dispatching costs, and linear service rewards, we prove optimality and monotonicity of threshold scheduling policies.     

Optimal control of polling models for transportation applications

We formulate and analyze a dynamic scheduling problem for a class of transportation systems in a Markov Decision Process (MDP) framework. A transportation system is represented by a polling model consisting of a number of stations and a server with switch-over costs and constraints on its movement (the model we have analyzed is intended to emulate key features of an elevator system). Customers request service in order to be transported by the server from various arrival stations to a common destination station. The objective is to minimize a cost criterion that incorporates waiting costs at the arrival stations. Two versions of the basic problem are considered and structural properties of the optimal policy in each case are derived. It is shown that optimal scheduling policies are characterized by switching functions dependent on state information consisting of queue lengths formed at the arrival stations.     

Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers

We consider a single server queueing system in which service shuts down when no customers are present, and is resumed when the queue length reaches a given critical length. We assume customers are heterogeneous on delay sensitivity and analyze customers’ strategic response to this mechanism and compare it to the overall optimal behavior. We provide algorithms to compute the equilibrium arrival rates and also derive the monotonicity of equilibrium and optimal arrival rates. We show that there may exist multiple equilibria in such a system and the optimal arrival rate may be larger or smaller than the decentralized equilibrium one.     

Towards an Erlang formula for multiclass networks

Consider a multiclass stochastic network with state-dependent service rates and arrival rates describing bandwidth-sharing mechanisms as well as admission control and/or load balancing schemes. Given Poisson arrival and exponential service requirements, the number of customers in the network evolves as a multi-dimensional birth-and-death process on a finite subset of ℕ ^k . We assume that the death (i.e., service) rates and the birth rates depending on the whole state of the system satisfy a local balance condition. This makes the resulting network a Whittle network, and the stochastic process describing the state of the network is reversible with an explicit stationary distribution that is in fact insensitive to the service time distribution. Given routing constraints, we are interested in the optimal performance of such networks. We first construct bounds for generic performance criteria that can be evaluated using recursive procedures, these bounds being attained in the case of a unique arrival process. We then study the case of several arrival processes, focusing in particular on the instance with admission control only. Building on convexity properties, we characterize the optimal policy, and give criteria on the service rates for which our bounds are again attained.